Question: Factor the quadratic expression completely. $-7x^2-24x-9=$
Solution: Since the terms in the expression do not share a common monomial factor and the coefficient on the leading $x^2$ term is not $1$, let's factor by grouping. The expression ${-7}x^2{-24}x{-9}$ is in the form ${A}x^2+{B}x+{C}$. First, we need to find two integers ${a}$ and ${b}$ such that: $\begin{cases} &{a}+{b}={B}={-24} \\\\ &{ab}={A}{C}= ({-7})({-9})=63 \end{cases}$ We find that ${a}={-3}$ and ${b}={-21}$ satisfy these conditions, since ${-3}+({-21})={-24}$ and $({-3})({-21})=63$. Next, we can use these values to rewrite the $x$ -term and factor by grouping. $\begin{aligned} -7x^2-24x-9&=-7x^2{-21}x{-3}x-9 \\\\ &=-7x(x+3)-3(x+3) \\\\ &=(x+3)(-7x-3) \end{aligned}$ In conclusion, $-7x^2-24x-9=(x+3)(-7x-3)$